(1+1/4*x^2)^(1/2)

Input

sqrt(1 + 1/4 x^2)

Result

sqrt(x^2/4 + 1)

Plot

Plot
Plot

Alternate form

sqrt(x^2 + 4)/2

Complex roots

x = -2 i
x = 2 i

Roots in the complex plane

Roots in the complex plane

Properties as a real function

Domain

R (all real numbers)

Range

{y element R : y>=1}

Parity

even

Series expansion at x=0

1 + x^2/8 - x^4/128 + O(x^5) (Taylor series)
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Series expansion at x=-2 i

(sqrt(2 - i x) sqrt(x + 2 i))/sqrt(x + 2 i) + (i sqrt(2 - i x) (x + 2 i)^(3/2))/(8 sqrt(x + 2 i)) + (sqrt(2 - i x) (x + 2 i)^(5/2))/(128 sqrt(x + 2 i)) + O((x + 2 i)^(7/2)) (generalized Puiseux series)
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Series expansion at x=2 i

(sqrt(2 + i x) sqrt(x - 2 i))/sqrt(x - 2 i) - (i sqrt(2 + i x) (x - 2 i)^(3/2))/(8 sqrt(x - 2 i)) + (sqrt(2 + i x) (x - 2 i)^(5/2))/(128 sqrt(x - 2 i)) + O((x - 2 i)^(7/2)) (generalized Puiseux series)
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Series expansion at x=∞

x/2 + 1/x - (1/x)^3 + O((1/x)^4) (Laurent series)
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Derivative

d/dx(sqrt(1 + x^2/4)) = x/(2 sqrt(x^2 + 4))

Indefinite integral

integral sqrt(1 + x^2/4)dx = 1/4 sqrt(x^2 + 4) x + tanh^(-1)(x/sqrt(x^2 + 4)) + constant

Global minimum

min{sqrt(1 + x^2/4)} = 1 at x = 0

Series representation

sqrt(1 + x^2/4) = sqrt(x^2/4) sum_(k=0)^∞ 4^k (x^2)^(-k) binomial(1/2, k) for abs(x)>2
sqrt(1 + x^2/4) = sum_(k=0)^∞ ((-1/4)^k (x^2)^k (-1/2)_k)/(k!) for abs(x)^2<4
sqrt(1 + x^2/4) = sqrt(x^2/4) sum_(k=0)^∞ ((-4)^k (x^2)^(-k) (-1/2)_k)/(k!) for abs(x)>2
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Integral representation

(1 + z)^a = ( integral_(-i ∞ + γ)^(i ∞ + γ) (Γ(s) Γ(-a - s))/z^s ds)/((2 π i) Γ(-a)) for (0<γ<-Re(a) and abs(arg(z))<π)
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